Question

In Problems \(67-70\), determine the domain of the variable \(x\) in each expression. \(\frac{4}{x-5}\)

Short answer

All real numbers except \(x = 5\).

Step-by-step solution

- Identify the restriction

For the expression \(\frac{4}{x-5}\), the denominator cannot be zero because division by zero is undefined.

- Set the denominator equal to zero

Set the denominator equal to zero to identify the value that causes the restriction: \(x - 5 = 0\).

- Solve for the restricted value

Solve the equation \(x - 5 = 0\) to find the restricted value. Adding 5 to both sides results in \(x = 5\).

- State the domain

The domain of \(\frac{4}{x-5}\) includes all real numbers except the restricted value. Thus, the domain is all real numbers except \(x = 5\).

Key concepts

headline of the respective core concept

Rational functions are fractions that have polynomials in the numerator and the denominator. The general form of a rational function is \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not equal to zero. One of the primary concerns in evaluating and working with rational functions is ensuring that the denominator is never zero. This would make the function undefined. Therefore, identifying points where the denominator is zero helps determine the function's domain. This is evident in the given exercise, where \(\frac{4}{x-5}\) is a rational function and determining its domain involves finding when the denominator \(x-5\) is zero.

undefined values

Undefined values in a function occur where the function does not produce a valid output. For rational functions like \(\frac{4}{x-5}\), an undefined value happens when the denominator equals zero. In this example, setting the denominator \(x-5\) to zero gives us the equation \(x - 5 = 0\). Solving for \(x\), we find \(x = 5\) is the value that makes the denominator zero. Thus, the function \(\frac{4}{x-5}\) is undefined at \(x = 5\). So, \(x = 5\) is excluded from the domain, because division by zero is undefined. Always check for these restrictions while dealing with rational functions to avoid undefined values in your solutions.

real numbers

Real numbers include all the numbers on the number line. This group encompasses both rational and irrational numbers. When determining the domain of a function, we typically look at all real numbers and then exclude any values that make the function undefined, such as those that make the denominator zero in a rational function. In the given exercise, \(\frac{4}{x-5}\), we exclude \(x = 5\) because it makes the denominator zero. Therefore, the domain of the function is all real numbers except 5. This is written as: \(\text{Domain: } \{x \in \mathbb{R} \mid x \eq 5\}\). It simply means all real numbers where \(x\) is not equal to 5, ensuring the function remains valid and defined.